On the lattice structure of pseudorandom numbers generated over arbitrary finite fields

Marsaglia's lattice test for congruential pseudorandom number generators module a prime is extended to a test for generators over arbitrary finite fields. A congruential generator eta (0),eta (1),...,generator by eta (n) = g(n), n = 0, 1,..., passes Marsaglia's s-dimensional lattice test if and only if s less than or equal to deg(g). It is investigated how far this condition holds true for polynomials over arbitrary finite fields F-q, particularly for polynomials of the form g(d) (x) = alpha (x + beta)(d) + gamma, alpha, beta, gamma is an element of F-q, alpha not equal 0, 1 < d < q - 1.